Papers

Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erdős-Heilbronn conjecture. In this note, we develop Das's linear algebraic method and give a new elementary proof of the Alon-Nathanson-Ruzsa theorem for restricted sumsets, which implies the Erdős-Heilbronn conjecture. Compared with the classical polynomial method via Combinatorial Nullstellensatz, our proof uses only basic linear algebra over finite fields, including Vandermonde matrices and solvability of linear systems.

Caetano et al. (Proc. R. Soc. A. 481:20230650, 2025) have proposed a formulation for sound-soft acoustic scattering by a compact scatterer O $\subset$ Rn, in which the scattered field is represented as an acoustic Newtonian potential whose density is the solution of an operator equation on a compact set $\Gamma$ $\subset$ O. In the case that $\Gamma$ is Ahlfors-David d-regular (a d-set), for some d $\in$ (n–2, n], they show, moreover, that the operator equation can be interpreted as an integral equation, the integration with respect to d-dimensional Hausdorff measure, and present a convergent Galerkin scheme for numerical computation. In this paper we make a substantial extension of these results so that they apply to more realistic fractal scatterers that are multifractal, in the sense that they have spatially varying fractal dimension. Firstly, we provide, inspired by Claret et al. (J. Math. Pures Appl. 212:103888, 2026), an interpretation of this operator equation as an equation between a trace space on $\Gamma$ and its dual, and, in many cases, relate the density to a notion of the normal derivative of the scattered field on $\Gamma$. Secondly, we show that the operator equation is equivalent to an integral equation on $\Gamma$ whenever $\Gamma$ is the support of a Radon measure $\mu$ such that: (i) the trace operator from H1(Rn) to L2($\Gamma$, $\mu$) is continuous and; (ii) certain canonical singular integrals with respect to $\mu$ are finite; and we characterise a large class of measures for which (i) and (ii) hold. Finally, we show that Galerkin methods based on finite element subspaces of L2($\Gamma$, $\mu$) are convergent if and only if, additionally, C$\infty$\_0 (Rn\$\Gamma$) is dense in the kernel of the trace operator. These results apply, in particular, if $\Gamma$ is a finite union of d-sets with different values of d. In the case that each d-set is the attractor of an iterated function system of contracting similarities, we establish rates of convergence for the Galerkin method.

A finite word $w$ is called closed if it has length at most 1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences in $w$. An infinite word $u$ is called closed-rich if the infimum of all possible ratios between the number of closed factors within any factor $w$ of $u$ and square of the length of $w$ exists and is positive. We define this infimum as the closed-rich constant $C_u$ of the infinite closed-rich word $u$. Puzynina and Parshina (2024) proved that infinite closed-rich words exist. In this paper, we study possible values of closed-rich constants of infinite closed-rich words. In particular, we estimate the supremum $C_{sup}$ of the closed-rich constants of infinite closed-rich words: we show that $C_{sup} \leq 0.165952$. Besides that, we study the closed-rich constant $C_f$ of the Fibonacci word $f$ and show that $ 0.09519 \leq C_f\leq 0.10893 $. In particular, this gives a lower bound for $C_{sup}$: $ 0.09519 \leq C_{sup}$.

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound in the remaining range. We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every $n\ge d+1$, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We determine sharp canonical stability in even diameter, sharp canonical and global stability in odd diameter, and prove a nontrivial general upper bound for global stability in even diameter. In particular, these two notions exhibit a sharp parity split: in odd diameter they collapse to the same problem, whereas in even diameter they lead to genuinely different extremal behavior.

We prove the existence of global solutions for some coupled systems of partially nonautonomous evolution inclusions comprised of a Cauchy problem with a compact resolvent semigroup generator and an evolution equation governed by a subdifferential of a real potential. Our system in particular includes nonautonomous generalized Schrödinger-Debye systems of inclusions with variable exponents, but extends to hyperbolic-parabolic systems of inclusions in particular to Maxwell-parabolic systems of inclusions. Methodologically, we extend an approach of Vrabie et al. to the nonautonomous case and make use of standard semigroup tools to accomodate non-parabolic behaviour of solutions paired with a new existence result for measurable selections. The combination of the latter two requires the set-valued coupling terms to be Hausdorff-continuous, to take bounded, convex and closed values, and to satisfy weak continuity with respect to one variable.

We prove two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first one resolves the \(p\equiv1\pmod4\) part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor \([\chi(j-k+1)]_{0\le j,k<(p-1)/2}\). The core evaluation is \[ \det H=\leg{2}{p}(b'_p-a'_p),\qquad U^TH^{-1}U= \begin{pmatrix} \leg{2}{p}\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\[2mm] \dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \end{pmatrix}, \] where \(U=(\mathbf1,\eta)\) and \(\eta_j=\chi(j)\). The proof uses Vsemirnov's factorisation of Chapman's matrix and an adjacent cofactor calculation. The second result gives a uniform exact congruence modulo \(p\) for the determinant underlying Sun's Conjecture 4.10(i), valid for any ordered half-system modulo sign and all \(u,v\in\mathbb F_p\). Its standard specialization recovers the asserted square class. The square-class assertion itself also follows from Sun's earlier evaluation of \(T(d,p)\); the contribution here is an exact and half-system refinement.

Motivated by the "composition theorems" of Chernikov-Hempel and Abd Aldaim-Conant-Terry we introduce $k$-trace definability between first order theories. Any theory which is $k$-trace definable in a NIP theory is $k$-NIP and any theory which is $2$-trace definable in a stable theory is $2$-NFOP. All known examples of $k$-NIP theories are $k$-trace definable in NIP theories. We show that for several of the main examples of $k$-NIP theories $T$ there is a NIP theory $T^*$ such that $T$ is the (unique up to a certain notion of equivalence) universal theory which is $k$-trace definable in $T^*$. For example the theory of Hilbert space is the universal theory which is $2$-trace definable in RCF, the theory of the generic class $k$ nilpotent Lie algebra over $\mathbb{F}_p$ is the universal theory which is $k$-trace definable in the theory of infinite $\mathbb{F}_p$-vector spaces, the theory of the generic $k$-hypergraph is the universal theory which is $k$-trace definable in the theory of a set with two elements, and the theory of Uryshon space is the universal theory which is $2$-trace definable in the theory of $(\mathbb{R}; +, <)$. We construct the universal theory $D_k(T)$ which is $k$-trace definable in an arbitrary theory $T$.

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of “very good” characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image.

Let $k$ be a positive integer. A graph is said to be $(P_2 \cup kP_1)$-free if it does not contain $P_2 \cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian or the Petersen graph. Note that this problem is affirmative for $k \in \{1,2,3\}$ by the known results. In this paper, we show that for each integer $k \geq 4$, if $G$ is a $1$-tough and $(k-1)$-connected $(P_2 \cup kP_1)$-free graph with $|V(G)| \ge k^2+k+1$ and $\delta(G) \ge k$, then $G$ is hamiltonian. This result implies that the above question is affirmative for large graphs.

It is well known that distributions whose symmetrized gradient is a bounded Radon measure belong to the space $BD$ on bounded domains with $\mathcal{C}^1$ boundary. In this work, we extend this result to a broader class of first-order linear elliptic operators. More precisely, let $\mathcal{A}$ be a first-order linear elliptic operator satisfying the rank-one property. We prove that if a distribution defined on a Lipschitz domain has bounded $\mathcal{A}$-variation, then it belongs to the space $BV^{\mathcal{A}}$.

Sun proposed a list of congruence and quadratic-residue conjectures for determinants and permanents over residue classes modulo a prime. This article gives a uniform treatment of Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list, while making explicit the overlap with two earlier contributions. Luo and Xia's Legendre-symbol formula for $D_p(b,1)$ already implies the non-vanishing assertion in Conjecture 4.6 when $p\equiv5\pmod {24}$; our determinant argument gives a root-quotient criterion for irreducible binary quadratic forms over $\Fp$ and also covers the remaining case $p\equiv19\pmod {24}$. For the Cauchy kernel $1/(x-y)$, we prove the derangement determinant and permanent congruences modulo $p^2$ and a polynomial fixed-point permanent congruence modulo $p$. For the Cayley kernel $(x+y)/(x-y)$, She, Sun and Xia's permanent identity supplies the structural input for the fixed-point permanent; combined with our Cauchy permanent congruence and Morley's congruence, it yields the congruence modulo $p^2$. Independent interpolation arguments give the signed fixed-point determinant congruences and the quadratic-residue assertion for the signed derangement determinant. Finally, a local expansion at the unique zero eigenvalue proves the half-size quadratic Cayley determinant divisibility by $p^2$, and by $p^3$ when $p\equiv7\pmod8$.

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the Hölder scaling condition. Our approach is based on a decomposition of the bilinear operator into square functions associated with linear cone multipliers and their variants. We derive pointwise bounds for these square functions via suitable strong maximal function estimates, and obtain sharp $L^4$ bounds using geometric methods originating in the work of Córdoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.

Let $\Gamma=(V,E)$ be a graph. The disjunctive domination number of $\Gamma$ is the minimum cardinality of a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$, $C_4\Box C_{n}$ and $C_8\Box C_{4n}$.

In this paper, we propose certain assumptions on the principal curvatures for a closed minimal hypersurface $M^5$ in $\mathbf{S}^6$ to be isoparametric, provided that the functions $S, f_3,f_4$ are constants. Our result removes the nonnegative scalar curvature assumption as in Tang and Yan \cite{TY}. Finally, as a rigidity result, if $M^5\subset \mathbf{S}^6$ has a point with exactly two distinct principal curvatures, then it must be a Clifford torus.

We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and fixing its value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder. In the high-energy regime this map is a twist map generated by a diagonally periodic generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c=0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather's converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni's theorem, this yields positive topological entropy for sufficiently small $c\geq0$. Our result improves previous similar results obtained via the standard Mather's converse-KAM criterion by giving a sharper quantitative threshold for the destruction of invariant curves.

Driven by the ultra-high throughput requirements of 6G, wireless communications are migrating to centimeter wave (cmWave) bands to overcome the limitations of current spectral resources. Massive multiple-input multiple-output (MIMO) and orthogonal frequency division multiplexing (OFDM) systems aim to achieve high spectral efficiency in cmWave regimes but are often constrained by the heavy overhead of downlink channel state information (CSI) feedback. This paper proposes a deep learning scheme based on the multi-axis multi-layer perceptron for image processing (MAXIM) architecture for joint semantic CSI feedback and hybrid beamforming in multi-user cmWave MIMO-OFDM systems, which maximizes the downlink sum rate by end-to-end optimization. Specifically, distributed encoders at multiple user equipments (UEs) perform limited CSI feedback, while the decoder at the base station (BS) jointly designs the hybrid beamforming matrices without explicit CSI reconstruction. The uplink transmission is implemented via deep joint source-channel coding (DJSCC) to enhance CSI compression efficiency and noise robustness. Furthermore, considering the high correlation between vertical and horizontal polarization channels in dual-polarized massive MIMO systems, a cross-polarization interaction module is introduced at the UEs to exploit polarization correlations for joint CSI compression. Simulation results demonstrate that the proposed method improves the downlink sum rate under various signal-to-noise ratio (SNR) conditions with a limited number of feedback symbols, validating its robustness and superiority in multi-user dual-polarized cmWave MIMO-OFDM systems.

The wakeup problem in distributed computing asks for a symmetric protocol that enables one of several processors to eventually guarantee that all (or, in a more general setting, enough) other processors have acted, using a shared register but no global clock. Dropping the symmetry requirement gives a well-known exercise often phrased in terms of prisoners entering, in an unknown sequence, a room equipped with a single binary switch, and using it to communicate. Kane and Kominers recently analysed a more general version of the latter with multiple parallel and indistinguishable rooms. We answer some open questions of Kane and Kominers regarding the minimum number of switch states needed for the prisoners to solve the problem. We also consider the symmetric “wakeup” version of this scenario, and establish exactly for which numbers of processors and registers a solution is possible.

We construct a surjective homomorphism from the (suitably interpreted) double loop-nilpotent $K$-theoretic Hall algebra to the Coulomb branch algebra of a quiver gauge theory, using the shuffle algebra interpretation.

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2} \] with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\leg{d}{p}$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p$. The required rank defect follows from a coefficient cancellation obtained from the involution $t\mapsto d/t$ on $\Fp^\times$ and the condition $\leg{d}{p}=-1$.

The finite families of Hahn polynomials and associated biorthogonal rational functions are interpreted algebraically in the framework of Leonard trios. We introduce the trio Hahn algebra and prove that it is isomorphic to the meta Hahn algebra, thereby clarifying the structural connection between Leonard trios and meta algebras. Finite dimensional realizations in terms of difference operators are constructed, and the functions of interest arise as overlaps between eigensolutions of ordinary eigenvalue problems. Their bispectral and biorthogonality properties follow naturally from the algebraic framework.

We propose a discrete privacy mechanism exploiting beneficial properties of the novel privacy measure Pointwise Maximal Leakage (PML). Given the utility assignment characterized by every input-output letter pair, we study the mechanism design problem that satisfies PML privacy guarantees and maximizes the worst-case utility. Unlike popular privacy measures like Differential Privacy (DP), PML allows us to set some conditional probabilities in the mechanism to be zero and thereby preventing the occurrence of some low utilities while preserving a strict PML constraint. We show that the utility-safe mechanism, with low computational complexity, is optimal for the worst-case utility problem with an additional constraint on the output support set. We finally demonstrate the effectiveness in several numerical experiments. Due to DP's inability to have zeros in the mechanism, the design of privacy mechanisms that optimize the worst-case utility is underexplored, and this work shows that PML is a privacy measure that is perfectly suited for this purpose.

Motivated by an inertial primal-dual dynamical system with vanishing damping, we propose a class of accelerated augmented Lagrangian methods with Nesterov extrapolation parameters for a linearly constrained convex optimization problem with a differentiable objective function. The framework contains two variants: an implicit-gradient scheme for convex continuously differentiable objectives and a partially explicit scheme for convex smooth objectives. Under suitable parameter conditions, we prove convergence of the primal-dual sequence to a primal-dual solution, together with accelerated estimates for the augmented Lagrangian gap, the feasibility violation, and the objective residual. In the noncritical parameter regime, these estimates are improved from $\mathcal{O}(1/k^2)$ to $o(1/k^2)$. Numerical experiments are also presented to illustrate the theoretical results. To the best of our knowledge, neither $o(1/k^2)$ rates for both feasibility violation and objective residual nor convergence of iterates under the critical parameter condition have been previously established for accelerated augmented Lagrangian-type methods in this setting.

We study the conjecture that a sum of the (g-1)st powers of adjoint Reidemeister torsions for a torus knot is an integer. We prove that the conjecture is true for any torus knot and all non-negative g. To prove the conjecture, we introduce the Verlinde numbers for torus knots from the viewpoint of modular S-matrix and show the recursion formulas and initial values of them. The recursion formulas of Verlinde numbers prove the integrality of the sum of the (g-1)st powers of adjoint Reidemeister torsions. Related to a modular S-matrix, we also provide a birational model of the character variety for a torus knot and show how to recover the adjoint Reidemeister torsion for a torus knot from the Hessian of the polynomial defining the birational model.

We derive a primal discontinuous Galerkin (DG) formulation for heterogeneous and anisotropic diffusion, obtained by exact algebraic elimination of the skeletal unknown in a compact hybridized interior penalty (H-IP) method. The resulting Unified Interior Penalty DG (UIP-DG) scheme involves transmissibility-based weights inherited from the hybrid formulation, together with two stabilization terms acting respectively on the primal jump and on the jump of the normal diffusive flux. These penalties scale, respectively, with the harmonic mean and with the inverse arithmetic mean of the face-wise transmissibilities. This construction provides a unified perspective on several interior penalty approaches previously introduced independently, while yielding a robust method with stability properties independent of the diffusion contrast and anisotropy. We prove consistency, coercivity, and boundedness of the formulation, and derive quasi-optimal energy-norm a priori error estimates for all variants. Numerical experiments confirm the theoretical claims.

We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction $T(\mathcal R)$, the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition ${\rm Tr}(T(\mathcal R))=1$ is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to $T(\mathcal R)$, and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information $g{\rm I}(\mathcal{G})_\rho$ associated with a chordal graph $\mathcal{G}$ and show that it is a relative-entropy discrepancy from $\rho$ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.

The number of finite additive 2-bases is known to grow exponentially. While this fact has been established by Marzuola and Miller (2010) using complex analytic techniques embedded in the study of numerical sets, we provide a direct, short proof using elementary probabilistic arguments.

The paper is a continuation of the paper of Kudla-Rapoport-Zink on $p$-adic uniformization of Shimura curves associated to a group of binary unitary similitudes. Here we consider two variants: first, the RSZ variant, for which we can allow any level which is maximal at the chosen special $p$-adic place where the group is anisotropic; second, the unitary group variant. The latter is based on an explicit determination of the integral local Shimura variety associated to an anisotropic unitary group over a $p$-adic local field.

This study investigates the impact of finite element selection on structural topology optimization using the SIMP (Solid Isotropic Material with Penalization) method. Specifically, it compares linear (P1) and quadratic (P2) triangular elements with the conventional bi-linear quadrilateral (Q1) elements. Numerical experiments performed on benchmark problems including a cantilever beam, a bridge structure, and a beveled beam reveal notable differences in both the final optimized objective value (compliance) and the accuracy of the finite element solutions. The accuracy is evaluated using an a posteriori error estimator, highlighting the influence of element type on solution quality and optimization performance.

We consider vector-valued magnetic Schrödinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries belong to $L^1_{\mathrm{loc}}(\mathbb{R}^d)$. We prove maximal inequalities in $L^p(\mathbb{R}^d;\mathbb{C}^m)$, $p\in[1,\infty)$ and the boundedness of the Riesz transforms $(\nabla - i\bm a)(-\bm \Delta_{\bm a}+V)^{-\frac{1}{2}}$ and $V^{\alpha}(-\bm \Delta_{\bm a}+V)^{-\alpha}$ on $L^p(\mathbb{R}^d;\mathbb{C}^m)$ for every $p \in (1,2]$ and every $\alpha\in[0,1/p]$.

Let $C\subset \mathbb P^3$ be a curve over an algebraically closed field of characteristic zero, and let $M(C)$ denote its Hartshorne-Rao module. We study how the geometry of $C$ influences whether $M(C)$ satisfies the Weak and Strong Lefschetz Properties. We first consider unions of general skew lines and prove that multiplication by $L^i$, for a general linear form $L$, has maximal rank on $M(C)$ for $i=1,2,3$. The proof uses a specialization to zero-dimensional schemes that can be written as a union of curvilinear schemes, each of a particular type and of degree at most three, together with generic Hilbert function results for such schemes, which are of independent interest. We then examine how special geometric configurations can affect the Weak Lefschetz Property. In particular, we show that curves on a smooth quadric surface have Hartshorne-Rao modules with the Weak Lefschetz Property, and that the property persists for unions of skew lines with all but one line on a quadric. By contrast, for $r\geq 10$, we construct configurations of $r$ skew lines with all but two lines on a quadric whose Hartshorne-Rao modules fail the Weak Lefschetz Property. Finally, we study smooth irreducible curves. We prove the Weak Lefschetz Property in several low-degree cases, construct a degree 15 curve for which it fails, and show that general nondegenerate rational curves have Hartshorne-Rao modules with the Weak Lefschetz Property. These results illustrate both the strength and the limitations of geometric hypotheses in controlling Lefschetz properties of Hartshorne-Rao modules.